Could a paradox kill an AI?
This classic problem exhibits a basic misunderstanding of what an artificial general intelligence would likely entail. First, consider this programmer's joke:

The programmer's wife couldn't take it anymore. Every discussion with her husband turned into an argument over semantics, picking over every piece of trivial detail. One day she sent him to the grocery store to pick up some eggs. On his way out the door,  she said, &quot;While you are there, pick up milk.&quot;
And he never returned.

It's a cute play on words, but it isn't terribly realistic.
You are assuming because AI is being executed by a computer, it must exhibit this same level of linear, unwavering pedantry outlined in this joke. But AI isn't simply some long-winded computer program hard-coded with enough if-statements and while-loops to account for every possible input and follow the prescribe results.
while (command not completed)
     find solution()

This would not be strong AI.
In any classic definition of artificial general intelligence, you are creating a system that mimics some form of cognition that exhibits problem solving and adaptive learning (←note this phrase here). I would suggest that any AI that could get stuck in such an &quot;infinite loop&quot; isn't a learning AI at all. It's just a buggy inference engine.
Essentially, you are endowing a program of currently-unreachable sophistication with an inability to postulate if there is a solution to a simple problem at all. I can just as easily say &quot;walk through that closed door&quot; or &quot;pick yourself up off the ground&quot; or even &quot;turn on that pencil&quot; — and present a similar conundrum.

&quot;Everything I say is false.&quot; — The Liar's Paradox